category of semigroups

 algebra
Notation SemiGrp\SemiGrp Objects semigroups, i.e. sets equipped with an associative binary operation Morphisms maps preserving the binary operation Related Grp\Grp, Mon\Mon, Rng\Rng External nLab Link

In contrast to monoids, semigroups do not need to have a neutral element, and in fact, they can be empty. This small difference has a huge impact on the categorical properties. For example, we do not have a zero object anymore.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: trivial semigroup
  • initial object: empty semigroup
  • products: direct products with pointwise operations
  • coproducts: similar to free products of monoids, except that the empty word is not allowed and the only condition on a word is that consecutive letters come from different factors (there are no neutral elements to exclude)

Special morphisms

  • isomorphisms: bijective morphisms
  • monomorphisms: injective morphisms
  • epimorphisms: A semigroup homomorphism f:T→Sf : T \to S is an epimorphism iff SS equals the dominion of U≔f(T)⊆SU \coloneqq f(T) \subseteq S, meaning that for every s∈Ss \in S we have s∈Us \in U or there are u1,…,um+1∈Uu_1,\dotsc,u_{m+1} \in U, v1,…,vm∈Uv_1,\dotsc,v_m \in U, x1,…,xm∈Sx_1,\dotsc,x_m \in S and y1,…,ym∈Sy_1,\dotsc,y_m \in S such that s=x1u1s = x_1 u_1, u1=v1y1u_1 = v_1 y_1, xi−1vi−1=xiuix_{i-1} v_{i-1} = x_i u_i, uiyi−1=viyiu_i y_{i-1} = v_i y_i, xmvm=um+1x_m v_m = u_{m+1} and um+1ym=su_{m+1} y_m = s.
  • regular monomorphisms:
  • regular epimorphisms: surjective morphisms